304 THE HANDBOOK OF PORTFOLIO MATHEMATICS
The answer returned by the objective function, along with the parame-
ters pumped into the objective function, gives us our coordinates at a certain
point inn+1 space. In the case of simply finding the optimalffor a single
market system or a single scenario spectrum,nis 1, so we are getting coor-
dinates in two-dimensional space. One of the coordinates is thefvalue sent
to the objective function, and the other coordinate is the value returned by
the objective function for thefvalue passed to it.
Since it is a little difficult for us to mentally picture any more than
three dimensions, we will think in terms of a value of 2 forn(thus, we
are dealing with the three-dimensional, i.e.,n+1, landscape). Since, for
simplicity’s sake, we are using a value of 2 forn, the objective function
gives us the height oraltitudein a three-dimensional landscape. We can
think of the north-south coordinates as corresponding to thefvalue asso-
ciated with one scenario spectrum, and the east-west coordinates as the
fvalue associated with another scenario spectrum. Each scenario spec-
trum pertains to the possible outcomes for a given market system. Thus,
we could say, for example, that the north-south coordinates pertain to the
fvalue for such-and-such a market under such-and-such a system, and
the east-west coordinates pertain to thefvalues of trading a different
market and/or a different system, when both market systems are traded
simultaneously.
The objective function gives us the altitude for a given set offvalues.
That is, the objective function gives us the altitude corresponding to a single
east-west coordinate and a single north-south coordinate. That is, a single
point where the length and depth are given by thefvalues we are pumping
into the objective function, and the height at that point is the value returned
by the objective function.
Once we have the coordinates for a single point (its length, depth, and
height), we need a search procedure, a mathematical optimization tech-
nique, to alter thefvalues being pumped into the objective function in such
a way so as to get us to the peak of the landscape as quickly and easily as
possible.
What we are doing is trying to map out the terrain in then+1-
dimensional landscape, because the coordinates corresponding to the peak
in that landscape give us the optimalfvalues to use for each market system.
Many mathematical optimization techniques have been worked out over
the years and many are quite elaborate and efficient. We have a number of
these techniques to choose from. The burning question for us is, “Upon
what objective function shall we apply these mathematical optimization
techniques?” under this new framework. The objective function is the heart
of this new framework in asset allocation, and we will discuss it and show
examples of how to use it before looking at optimization techniques.